That's a difficult one to achieve.

Rationality is about how to process evidence to change one's prior, it has very little to say about what belief you start with, besides the fact that it must be expressible with classical logic.
To complicate the matter, Bayesian evidence works in such a way that if you classify something as evidence, then it means that its absence will lower the probability of the assertion it is supporting.

To have a belief that is both rational and unsupported, you must start with a model that is at one time compatible with background information, whose support is difficult to obtain and is a better fit than competing models, who might even have easier to obtain evidence.
A tough challenge!